Mathematics provides the formal backbone for virtually every area of computer science. These notes cover the core mathematical topics a CS student encounters - from discrete structures and counting arguments to linear algebra and calculus - with an emphasis on how each area connects back to computation.
Discrete Math
The bread and butter of CS math: structures that are countable, finite, or combinatorial.
- Graph Theory - formal graph definitions, planarity, coloring, Euler and Hamilton paths
- Combinatorics - permutations, combinations, pigeonhole, inclusion-exclusion
- Relations and Equivalence - relations, their properties, and equivalence classes as partitions
- Functions: Injective, Surjective, Bijective - the three properties and what each guarantees
- The Pigeonhole Principle - the counting argument that proves the impossible impossible
- Boolean Algebra - the algebra of truth values and the laws behind circuit simplification
Linear Algebra
Vectors, matrices, and transformations that power graphics, machine learning, and scientific computing.
- Linear Algebra Fundamentals - vectors, matrices, transformations, eigenvalues
- Vectors and Dot Products - vectors as magnitude and direction, and what the dot product measures
- Matrices and Linear Transformations - a matrix as a map between spaces, rank, and composition
- Eigenvalues and Eigenvectors - the directions a transformation only stretches
- Singular Value Decomposition - rotate, scale, rotate, and the optimal low-rank approximation
Calculus
Continuous mathematics for analysis of algorithms, probability distributions, and optimization.
- Limits and Continuity - the limit as the foundation, and what continuity actually requires
- Derivatives and Gradients - instantaneous rate of change, and the gradient as the direction of steepest ascent
- Integrals and the Fundamental Theorem - the integral as accumulation, and the theorem tying it to differentiation
- Sequences and Series - convergence, geometric closed forms, and the sums that show up in algorithm analysis
- Taylor Series and Approximation - approximating a function by a polynomial built from its derivatives, and the remainder as an engineering contract
Growth and scale
- Logarithms and Exponentials - the inverse pair, change of base, and why the base vanishes from complexity results
Optimization
- Convexity and Optimization Basics - convex sets and functions, and why convexity makes a local minimum global
- Linear Programming and Duality - the feasible region as a polytope, simplex against interior-point, and what duality buys
Number Theory
Divisibility, primes, and modular arithmetic - the engine behind cryptography.
- Number Theory and Modular Arithmetic - divisibility, primes, congruences, and the arithmetic cryptography runs on
Logic & Proofs
Formal reasoning techniques that underpin verification, type theory, and specification.
- Propositional Logic - connectives, truth tables, tautology vs contradiction
- Predicate Logic and Quantifiers - predicates, quantifiers, nested-quantifier order
- Proof Techniques - direct, contrapositive, contradiction, cases, counterexample
- Mathematical Induction - weak, strong, and structural induction
- Set Theory Basics - sets, operations, subsets, power sets, cardinality
Combinatorics & Probability
Counting arguments and probabilistic reasoning for algorithm analysis and randomized methods.
- Combinatorics - permutations, combinations, pigeonhole, inclusion-exclusion
- Discrete Probability - sample spaces, Bayes’ theorem, expected value
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